The purpose of this research is to develop new statistical methods for analyzing health related studies with missing data when the probability of non-response depends on the subject's unobserved measurements, i.e. when non-response is non-ignorable. Non-ignorable non-response is suspected, for example, in clinical studies of the elderly in which cognitive impaired subjects may be less likely to want or be able to participate, in quality of life studies in which worse off patients are less likely to be able to complete assessment tests, and in studies of socially sensitive outcomes like drug consumption or sexual behavior. The validity and usefulness of currently available methods for the analysis of non-ignorable non-response critically depends on highly restrictive parametric modeling assumptions likely to be violated in many health related studies. This research will develop semiparametric methods that are valid under less restrictive modeling assumptions and hence applicable to a wide spectrum of studies. This will entail: (1) the investigation of the identifiability of parameters of interest under specific semiparametric models, (2) the derivation of a comprehensive inferential approach when the parameters of interest are identifiable under the partially specified semiparametric models, (3) the derivation of efficient semiparametric estimators that effectively extract the information available in the observed data given the knowledge encoded in the semiparametric model. The methods derived in this work will be useful for conducting semiparametric sensitivity analyses. This work will also investigate methods for analyzing studies with follow-up of a, possibly biased, sample of non-respondents. Currently available parametric models for the analysis of continuous non-ignorable outcomes have a distinctive property that complicates the analysis: they are identifiable but have non-invertible Fisher information matrix. The distributional properties of the maximum likelihood estimators and of the likelihood ratio and score tests in these models are presently unknown. This research will derive the distributional theory for likelihood based inference in parametric identifiable models with non- invertible Fisher information.